[SauronOfMordor]

When the moon eclipses the sun, it is directly in line between the earth and the sun, so the center of mass (and the center of gravitational attraction) will have its maximum shift sunward along a line stretching between the earth and the sun...and near which the observer of the eclipse will be.

If so, then the effect should also be observed whenever the moon is more-or-less in line with the sun (ie, once every month), rather than exclusively being seen during an eclipse

[aruanan]

If so, then the effect should also be observed whenever the moon is more-or-less in line with the sun (ie, once every month), rather than exclusively being seen during an eclipse

No, if there's shielding going on, the zone of totality would be the place where the effect would be the greatest.